Geometry 10.1 Tangents to Circles Good Afternoon! Please place your phones in the pockets up front and grab the computer that matches your desk number. Log into Canvas and click on Courses – Geometry; go to modules; Chapter 10; 10.1 day 1 warmup. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles 10.1 Warmup Day 1 Write in ascending order (from least to greatest). Show work. Evaluate the following: 2. 3 5 + 2 7 3. 4 7 ∙ 11 13 4. 2 3 −4 5. 6 5 ÷ 9 2 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles 10.1 Warmup Day 1 (Answers) Write in ascending order (from least to greatest). Show work. Evaluate the following: a. 3 5 + 2 7 b. 4 7 ∙ 11 13 c. 2 3 −4 d. 6 5 ÷ 9 2 7 11 , 2 3 , 23 33 a. 31 35 b. 44 91 c. − 10 3 d. 4 15 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles 10.1 Warmup Day 2 Identify the parts of the diagram 1. All tangents 5. Circle center 2. All points of tangencies 6. All radii 3. All secants 7. All diameters 4. All chords Solve for x. What kind of segment is 𝐶𝐴 ? 8. What kind of segment is 𝐵𝐶 ? 10 x December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry Circles Chapter 10 10.1 Lines and Segments that Intersect Circles Circles Chapter 10 mbhaub@mpsaz.org

Geometry 10.1 Tangents to Circles 10.1 Essential Question What relationships exist among two tangent segments or a radius and a tangent segment? December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Goals Know properties of circles. Identify special lines in a circle. Solve problems with special lines. December 1, 2018 Geometry 10.1 Tangents to Circles

Circle: Set of points on a plane equidistant from a point (center). B This is circle C, or C CR is a radius. AB is a diameter. R A The diameter is twice the radius. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Terminology One radius Two radii radii = ray-dee-eye December 1, 2018 Geometry 10.1 Tangents to Circles

All Radii in a circle are congruent December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Interior/Exterior A is in the interior of the circle. A C is on the circle. C B is in the exterior of the circle. B December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Congruent Circles Circles are congruent, if their radii are congruent. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Lines in a circle. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Chord A chord is a segment between two points on a circle. A diameter is a chord that passes through the center. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Secant A secant is a line that intersects a circle at two points. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles A tangent is a line that intersects a circle at only one point. It is called the point of tangency. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Example 1 Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of ⊙C. a. 𝑨𝑪 b. 𝑨𝑩 c. 𝑫𝑬 d. 𝑨𝑬 e. 𝑨𝑮 f. 𝑭𝑪 g. 𝑭𝑨 h. 𝑫𝑭 Radius Diameter Tangent Secant Chord Radius Secant Tangent December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Tangent Circles Intersect at exactly one point. These circles are externally tangent. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Tangent Circles Intersect at exactly one point. These circles are internally tangent. December 1, 2018 Geometry 10.1 Tangents to Circles

Can circles intersect at two points? YES! However, the circles are NOT tangent! December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Concentric Circles Have the same center, but different radii. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Concentric Circles Have the same center, different radius. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Concentric Circles Have the same center, different radius. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Concentric Circles Have the same center, different radius. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Concentric Circles Have the same center, different radius. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Concentric Circles Have the same center, different radius. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Concentric Circles Have the same center, different radius. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Concentric Circles Have the same center, different radius. December 1, 2018 Geometry 10.1 Tangents to Circles

Common External Tangents And this is a common external tangent. This is a common external tangent. December 1, 2018 Geometry 10.1 Tangents to Circles

Common External Tangents in a real application… December 1, 2018 Geometry 10.1 Tangents to Circles

Common Internal Tangents And this is a common internal tangent. This is a common internal tangent. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Example 2 Tell how many common internal tangents and common external tangents the circles have and draw them. 2 internal tangents external tangents 1 2 internal tangents external tangents 2 internal tangents external tangents These circles do not __________. These are ________ ________ circles. externally These circles ___________. intersect tangent intersect December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Your Turn Tell how many common tangents the circles have and draw them. State whether the tangents are external tangents or internal tangents. What additional name can be used to describe the circles in 4? 5? 2 internal tangents external tangents 1 internal tangents external tangents internal tangents external tangents Internally tangent circles Concentric circles December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Quick Quiz Match each part with the best term for that part. 1. 2. 3. 4. 5. 6. 7. 8. E G D B A C H F December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Theorems December 1, 2018 Geometry 10.1 Tangents to Circles

Theorem 10.1 (w/o proof) Radii are Perpendicular to Tangents If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. December 1, 2018 Geometry 10.1 Tangents to Circles

Theorem 10.2 (w/o proof) Tangents are Perpendicular to Radii If a line drawn to a circle is perpendicular to a radius, then the line is a tangent to the circle. (The converse of 10.1) December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Example 3 Is RA tangent to T? YES R 12 A 5 52 + 122 = 132 25 + 144 = 169 169 = 169 13 T TA = 13 RAT is a right triangle. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Example 4 Find the length of AC. C 82 + AC2 = 122 64 + AC2 = 144 AC2 = 80 AC = 4 5 A 8 B 12 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles FOIL Find (x + 3)2 (x + 3)(x + 3) December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles FOIL Find (x + 3)2 (x + 3)(x + 3) x2 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Review FOIL Find (x + 3)2 (x + 3)(x + 3) 3x x2 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles FOIL Find (x + 3)2 (x + 3)(x + 3) 3x x2 + 3x December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles FOIL Find (x + 3)2 (x + 3)(x + 3) 9 x2 + 3x + 3x December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles FOIL Find (x + 3)2 (x + 3)(x + 3) x2 + 3x + 3x + 9 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles FOIL (x + 3)2 = x2 + 6x + 9 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Expand (x + 9)2 (x + 9)(x + 9) F: x2 O: 9x I: 9x L: 81 (x + 9)2 = x2 + 18x + 81 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Example 5 BC is tangent to circle A at B. Find r. A B C r 16 24 D DC = 16 AC = r + 16 AC = ? r r2 + 242 = (r + 16)2 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Solve the equation. r2 + 242 = (r + 16)2 r2 + 576 = (r + 16)(r + 16) r2 + 576 = r2 + 16r + 16r + 256 576 = 32r + 256 320 = 32r r = 10 r2 + 242 = (r + 16)2 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Here’s where the situation is now. 26 A 10 D 16 10 C B 24 Check solution: 102 + 242 = 262 100 + 576 = 676 676 = 676 AC = 26 r = 10 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Theorem 10.3 If two segments from the same exterior point are tangent to a circle, then the segments are congruent. Theorem Demo December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Example 6 HE and HA are tangent to the circle. Solve for x. A 12x + 15 H 9x + 45 E December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Solution 12(10) + 15 120 + 15 = 135 12x + 15 = 9x + 45 3x + 15 = 45 3x = 30 x = 10 A 12x + 15 H 9x + 45 9(10) + 45 90 + 45 = 135 E December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Try This: The circle is tangent to each side of ABC. Find the perimeter of ABC. 7 + 12 + 9 = 28 A 2 2 9 7 7 5 C B 7 5 December 1, 2018 12 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Factoring Review December 1, 2018 Geometry 10.1 Tangents to Circles

1. Greatest Common Factor Look for something that each term has in common, and factor that out. x3 + x2 b. 2x3 – 6x c. 3x3 + 3x2 - 6x December 1, 2018 Geometry 10.1 Tangents to Circles

2. Difference of Two Squares Are both terms perfect squares and is there a subtraction between them? x2 – 9 b. 9x2 – 49 c. 3x2 – 12 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles 3. Trinomial: 𝑎 𝑥 2 +𝑏𝑥+𝑐 Find the factor pairs for c. Then choose the pair that adds to b. a. x2 + 2x − 8 b. x2 – 17x + 60 c. 2x2 + 12x + 16 d. 3x2 – 21x − 90 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Example 7 BC is tangent to circle A at B. Find x. A B C 9 x 12 D AC = ? AC = x + 9 9 AD = 9 AD = ? 92 + 122 = (x + 9)2 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Solve the equation. 92 + 122 = (x + 9)2 81 + 144 = (x + 9)(x + 9) 225 = x2 + 9x + 9x + 81 225 = x2 + 18x + 81 0 = x2 + 18x – 144 0 = (x – 6) (x + 24) Factors of –144 −1∙144 −2∙72 −3∙48 −4∙36 −6∙24 −8∙18 −9∙16 −12∙12 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Solve the equation. 0 = (x – 6) (x + 24) So… x – 6 = 0 or x = 6 or x + 24 = 0 x = -24 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Here’s where the situation is now. If x = 6, then… 15 A 9 D x = 6 9 92 + 122 = 152 81 + 144 = 225 225 = 225 C B 12 However, if x = -24, then AC = -15. Since the side lengths can’t be negative, x = -24 is not a solution. December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Can you… Identify a radius or diameter? Recognize a tangent or secant? Define Concentric circles? Internally tangent of circles? Externally tangent of circles? Tell the difference between internal and external tangents? Solve problems using tangent properties? December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Practice Problem 1 MD and ME are tangent to the circle. Solve for x. D 4x – 12 = 2x + 12 2x – 12 = 12 2x = 24 x = 12 4x  12 M 2x + 12 E December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Practice Problem 2 R T 4 12 x Solve for x. x2 + 42 = (4 + 12)2 x2 + 16 = 256 x2 = 240 x = 4 15 ≈ 15.5 December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Practice Problem 3 R T 6 8 x Solve for x. x2 + 82 = (x + 6)2 x2 + 64 = x2 + 12x + 36 64 = 12x + 36 28 = 12x x = 2.333… December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Practice Problem 4 HE and HA are tangent to the circle. Solve for x. A 𝑥 2 +10𝑥 H 3𝑥−12 E December 1, 2018 Geometry 10.1 Tangents to Circles

Geometry 10.1 Tangents to Circles Homework mbhaub@mpsaz.org December 1, 2018 Geometry 10.1 Tangents to Circles